Hi, I'm having trouble with the mesh on this mug handle
What you're seeing there are faces with the normals (the vectors orthogonal to the surfaces of the faces) pointing into the mesh as opposed to out of it. Unlike reality, 3D modeling is all smoke and mirrors. In many environments faces are only visible from one side. Blender lets you see backfaces, but they appear darker because they face away from the "light" for viewport shading.

To fix the problem, go into edit mode, select your whole mesh with A, and hit CTRL+N to recalculate normals. With a simple shape like that Blender will be able to figure out which way they need to be pointing.

Otherwise, you can select the affected faces, and W > Flip Normals to manually flip them the other way around.